How to animate morphing a rectangular plane into a sphere
I think the one who made that reference GIF (in another software) used two Simple Deform modifiers:


Create a plane, switch to Edit mode, SY2. then use a loop cut to divide it into two squares, then subdivide;
Add a Simple Deform modifier, set deform method to Bend, use a new empty object as its origin, select the empty, then RX90. (to bend it along an expected Z axis)
Add another Simple Deform modifier, also set deform method to Bend, use a second empty object as the origin, select the empty, then RX90, RZ90.
At frame 1, set 0 as the Angle value for both Bend modifiers, insert keyframe for them. Then jump to, say, frame 60, set 180 for the first Angle, 360 for the second Angle (for 360, you may have to type on keyboard), again, intert keyframe for both.
If normal direction is incorrect, flip it in Edit Mode.








UPDATE: In the incoming v2.8, the Simple Deform modifer has a new option that allows choosing Local Axis, so you can use only one Empty in this case, like this: